To obtain \(a_{(4)}\) , one considers the map \(p: \operatorname{\mathfrak {gl}}(7) \longrightarrow \mathbb {R}^7\)  (already considered in [1]}, [2]}) defined by
\(p(a) = \textstyle \sum _{i\in Z_7} (a_{i+1 \, i+3} - a_{i+3 \, i+1} + a_{i+4 \, i+5} - a_{i+5 \, i+4} + a_{i+2 \, i+6} - a_{i+6 \, i+2}) e_i.\) 
